MathDB

G2

Part of 2019 IMO Shortlist

Problems(1)

MP = NQ wanted, incircles related

Source: IMO 2019 SL G2

9/22/2020
Let ABCABC be an acute-angled triangle and let D,ED, E, and FF be the feet of altitudes from A,BA, B, and CC to sides BC,CABC, CA, and ABAB, respectively. Denote by ωB\omega_B and ωC\omega_C the incircles of triangles BDFBDF and CDECDE, and let these circles be tangent to segments DFDF and DEDE at MM and NN, respectively. Let line MNMN meet circles ωB\omega_B and ωC\omega_C again at P≠MP \ne M and Q≠NQ \ne N, respectively. Prove that MP=NQMP = NQ.
(Vietnam)
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